Codes for Optical CDMA Report with matlab code for simulation
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(2) !" . c.
(3) c. . #. . $ . . . . . # $ . . % $. . ! . !. $ ! $ & . '. ( )
(4) *
(5) + (
(6) ) ,* + (
(7) ) . )..
(8). ( ( % ++*( ( (-( c. O á . !. .
(9) . . c c c. (-. . !.á.. - . ! . .. .) *) (
(10) "#$$ %. /01/2. c. O 3/40. c. &' $ ((. c. )
(11) *+ .
(12) 5$! c c c c c ,
(13) c - c c c c. c c
(14) c cc #
(15) c c
(16) c
(17) ,c c ,c c c c
(18) c!c cÔ c
(19) c ,ccc
(20) cc c c c .c
(21) c c - c c
(22) c c c c ,c c c c c c -c c c c c c
(23) c
(24) ,c c -c - c c , c c hc #
(25)
(26) c c ,c c c c
(27) c c c c c
(28) c c
(29)
(30) c
(31) ,c c ,c c c !c c c c c c .c cc c c c c c
(32) cc c c c c
(33) c c c c c. !" c c c c.
(34) Ô
(35)
(36)
(37)
(38)
(39)
(40) . c. w w
(41) ! "# #
(42) $ % "
(43) &'#
(44) (
(45) ')
(46) $* +w $
(47) ( #) w,- "
(48) #.# $ #
(49)
(50) "' ) / )
(51) ' 0 w &&%' # ' '/'
(52)
(53) '/! & 12*#'(% %'$
(54) % %(
(55) 3 ' c
(56). . c. . cc. . !. "# $$ %!. c. . c. Ô c c
(57) c c. c c c c. c.
(58) m
(59) . m . . ë. . ë. .
(60)
(61)
(62) . . ë.
(63)
(64) ë
(65)
(66) . . .
(67)
(68) . . .
(69)
(70) . . ë. ! . ". .
(71)
(72)
(73) . #. .
(74) $% . &. ".
(75)
(76) ' ( . ). &. . ëë. & ë. ' . ë . & . $* . ë . ). . ë). . . . .
(77) c ô ô. . . c c c c c
(78) c c c c c c c c
(79) c c c c c c cc c c c ccccccc c c
(80) c c c
(81)
(82) c c c c c
(83) c c !"#ccc cc$
(84) c c$ c. c c c
(85) %&
(86) cc
(87) c c c
(88) c c. 'c
(89) c c ccc cc c c c ccc. cccccccc&cc c ccc ccc c cc c c c &c c c c c c c
(90) c c 'c $
(91) c c $ c $($. c ) c $$) c
(92) c c c c c c
(93) c c &c
(94) c c *+c ",c -c c c c c $
(95) c $
(96) c c $$ c cc c c
(97) c cc cc&c ccc c
(98) ccc cc$(
(99) c$$c"( c$$ c c. "( c $$c c c & c c c c
(100) c c c . c %
(101) c c c. c c/cc c c c0cc ccccc cc ccc"1&2 c cc c ccc ccc c cc"( c$$cc
(102)
(103) ccc c c (c c c c c "341 2 c "34c 1 c 5c c c 4( c $$c &c c c & c c c
(104) c & c c c &c c c c c c 6c 31 2 c3c1 c c. 'cc%c cc
(105)
(106) c c& cc& c cc7c c"(c ccc&c. & c c c c
(107) c c& c8cc
(108)
(109) c c4( c$$ cc c& c cc7c c4(c c
(110) cc
(111)
(112) c cc c c c &c cc c c
(113) c c8c c cc
(114) &c
(115) c
(116) c c (
(117) cc (
(118) c c4( c$$c c c c. c. cc c.
(119) · c
(120) c
(121) c c ሼͲǡͳሽȂc c c
(122) c ݊ c 9 c c ݓc c &c c c c. ሺ݊ǡ ݓǡ ݇ሻc$
(123) c$
(124) c c$$ ܥccͳ ݇ ݓ ݊cc ି. ࢞ሺሻ࢟ሺ ْ ࣎ሻ ୀ. . ccc c cሼݔǡ ݕሽcc ܥcc ݕ ് ݔc cc ߬ ് Ͳc. c c c ْ c c c
(125) ݊ c ݇ c c c %c
(126)
(127) c. c : c c. 8Ȱሺ݊ǡ ݓǡ ݇ሻc c c
(128) c &
(129) c
(130) c c c ሺ݊ǡ ݓǡ ݇ሻc $
(131) c. $
(132) c c$$ c c · c!4c;c<c0# ' c ܥccሺ݊ǡ ݓǡ ݇ሻc"( c$$cc&c
(133) cc
(134) c c c c cc ܥcc. c cc c c cccሺ݊ǡ ݓǡ ݇ሻc c7ൌ ݊ȁܥȁ c-c c
(135)
(136) c. c c c= c& c ሺܬ ǡ ܬ ǡ ܬ ሻ c c c c-c& c>c "c = c& c >c!4c;#c ି. ି. 5 ሺǡ ࢝ǡ ሻ ࢝ ඌ࢝ି ǥ ǥ ǥ ቔ ࢝ିቕඐ ǣ ൌ ࡶ ሺǡ ࢝ǡ ሻǤ. 4c ' c= c& c5>c!<#: c ݓଶ ݊݇ c 5 ሺǡ ࢝ǡ ሻ ܖܑܕሺǡ ቔ. ࢝ି. ࢝ ି. ቕሻǣ ൌ ࡶ ሺǡ ࢝ǡ ሻǤ . ;c ' c= c& c>!0# c ିሺିሻ ି ǥ ǥ ǥቔ ࢎቕ ǥ ቕඐ ǣ ൌ ࢝ିሺିሻ ࢝ ࢝ି. 5 ሺǡ ࢝ǡ ሻ ඌ ቔ. c. ࡶ ሺǡ ࢝ǡ ሻǤ. cc c.
(137) ሺିሻሺ௪ିሻ. cF ൌ ቀ݊ െ ݈ǡ ቔሺ௪ିሻ మ ିሺିሻሺିሻቕቁǤc. ݈ c c c ͳ ݈ ݇ െ ͳǡc c c. ሺ ݓെ ݈ ሻଶ ሺ݊ െ ݈ ሻሺ݇ െ ݈ ሻǤc. ccߔ ሺ݊ǡ ݓǡ ݇ ሻccc
(138) c &
(139) c7c cc"( c$$cc$$c ܥc c7cܲ ccc c&cc 3c P ccܲ ൌ ߔ ሺ݊ǡ ݓǡ ݇ ሻccc . 3c ^ c݈݅݉՜ஶ ఃሺǡ௪ǡሻ ൌ ͳc. c 8 · c cc
(140) c
(141) c c c"( c$$cc$c cc c
(142) c c c?"c!3c4@c"3c"0c"Ac"/c4"c4;c4<c40# ccc?c4c!Ac43# c ccc
(143)
(144) c c c &c c c !/c @c 4/#c c c c
(145) c c c = c & c c c c c &c c= c& c c&ccc c
(146) cc= c& c5c!0#c .
(147) cc c c c c c ccccccccc ccc. c cc ccc cc c c c c cc . c c
(148) ccc cc c c&c c. cc%(
(149)
(150) cc : c݇ c
(151) c c"ccc ݓൌ ͷ ሺͷ െ ͳሻ݇ cc 0(" c c c %c &
(152) c c c &c c <c c c c c c c c c. c ݓൌ ͷ c c ݇ ൌ ͳǡc c = c & c c
(153) . ି. ି. 5ሺǡ ࢝ǡ ሻ ࢝ ඌ࢝ି ǥ ǥ ǥ ቔ࢝ିቕඐ ǣ ൌ ࡶ ሺǡ ࢝ǡ ሻǤ ି ቕඐ ࢝ ࢝ି. 5 ሺǡ ǡ ሻ ඌ ቔ. . െ ࢝ሺ࢝ െ ሻ. . c cc
(154)
(155) cc െ ͺݔͷݔͶ. $݊ ͳͳ c c c c (c c
(156) c
(157) c c % c · c c c
(158) c &
(159) c c
(160) c Bc c c c &
(161) c c
(162) cc (cc
(163)
(164) c
(165) cc
(166) c cccc
(167) c c cc. c. cc c.
(168) . · 5c cc& c
(169) cccc (c. c c&c c&
(170)
(171) c c3c1c 2 c c O
(172) -c c c.
(173) ( (
(174)
(175) % c c c ( c c.
(176) c c c c &
(177) c c c c & c
(178) c c c. !;<#. c -c. c c c
(179) c
(180)
(181) c
(182) (c c c c
(183)
(184) (
(185) c c9cc
(186)
(187) c c cc cc (c
(188) c$$c4( c$$ c c. c4( c ሺܹܶݔǡ ݓǡ ݇ሻc$$cccc
(189) c c ሼͲǡͳሽሺܹܶݔሻcc c c cc. Bccሼܣǡ ܤሽ c ccc ܥcc c c >c ࢃ ࢀି. ሺࣅǡ ࢚ሻሺࣅǡ ሺ࣎ ࢀ ۩ܜሻሻ . ࣅୀ ࢚ୀ. c. cc ܣൌ ܤc ݐൌ Ͳc c
(190)
(191) c c c ݇ ccc%c
(192)
(193) cc. : cC cc ccc
(194) c
(195) ccc%c c. 'c c &c c c c c &
(196) c c c c 4( c ሺܹ ܶݔൌ ݔǡ ݓൌ ͷǡ ݇ ൌ ͳሻc. $$c c7c/c. c&
(197) c c cc4( c$$c-cc ccc
(198) c"( c $$c c& c B%
(199) c c cc (cc % c c "34c 1 c cc 4( c c c c c c c &c c c c c c c c c. c. cc c.
(200) ¬c. ^
(201)
(202)
(203)
(204)
(205)
(206) ^ O . . cons raons fo
(207)
(208) owng resrcons on e p
(209) acemen of pu
(210) ses w n an. Pracca
(211)
(212). array.And ermed em as> 3c Arrays w À
(213) each row of every ሺܹ array n. ሻ code. s requred o have Hammng wegh = 1.. 3c Arrays wh À À
(214) here each row of any ሺܹ. ሻ code n. s requred o have Hammng wegh < 1.. 3c Arrays wh À
(215) À ! here each co
(216) umn of every ሺܹ code array n. ሻ. s requred o have Hammng wegh = 1.. 3c Arrays wh À À
(217) À ! here each co
(218) umn of any ሺܹ. c. ሻ array n. s requred o have Hammng wegh < 1.. c cc.
(219) O ÀÀÀ The opca
(220) corre
(221) aor can be mp
(222) emened as shown n hs fgure.. ¬c. ^ ^
(223)
(224) 3c ef 4 x 4 marx represens a 2-D OO. 3c 4 recangu
(225) ar b
(226) ocks represens opca
(227) corre
(228) aor. 3c 9gh sde s receved oupu me a
(229) gned Each recangu
(230) ar box represens an opca
(231) f
(232) er mp
(233) emened usng a fber-5ragg grang or an opca
(234) mcro-resonaor. Ths f
(235) er ref
(236) ecs
(237) gh of he wave
(238) engh shown a
(239) ongsde he box and a
(240)
(241) ows
(242) gh of a
(243)
(244) oher wave
(245) enghs o pass hrough. F
(246) ers p
(247) aced furher a
(248) ong he ref
(249) econ pah w
(250)
(251) suffer an ncreased de
(252) ay and n hs manner, he p
(253) acemen of he f
(254) ers can be ad6used o brng he pu
(255) ses of a
(256)
(257) he dfferen wave
(258) enghs n he desred code marx no me a
(259) gnmen a he oupu of he corre
(260) aor. . c. cc c.
(261) O 8 · "# $8O . c c c c c
(262) c c ܹǡ ܶǡ ݓǡ ݇ǡ c
(263) c ߔ ሺܹܶݔǡ ݓǡ ݇ሻ c c c
(264) c &
(265) c.
(266) c ccሺܹǡ ܶǡ ݓǡ ݇ሻ cc4( c$$c c c c
(267) cc
(268)
(269) c c. 4( c$$ccc ccc c c"( c$$cc c4"c cc c c O 8
(270) ' c ܥc c c ሺܹܶݔǡ ݓǡ ݇ሻc 4( c $$c c &c
(271) c c
(272) (
(273) c c c c. cc ܥc c c cc c c c cc cc cc.
(274) c cc ܹ ܶݔcc c cc "( c c c
(275) c ܹܶ c-c
(276) c c c c c c ܹܶǡ ݓǡ ݇ሻc c 7c ?ܶȁܥȁc -c & c
(277)
(278) c c c
(279) c & c c c c c c& c c4( c$$cccc cc c4"c c. -c cc= c5 c c c4( c$$c. <c = c& c >c!;<#c ࢃ ࢃࢀି ࢃࢀି ǥ ǥ ǥ ቔ ࢝ି ቕඐ ǣ ൌ ࢝ି. 5 ሺࢃ࢞ࢀǡ ࢝ǡ ሻ ࢝ ඌ. ࡶ ሺࢃ࢞ࢀǡ ࢝ǡ ሻǤ. 0c ' c= c& c5>c!<#: c ݓଶ ݊݇ c 5ሺࢃ࢞ࢀǡ ࢝ǡ ሻ ܖܑܕሺࢃǡ ቔ. ࢃሺ࢝ିሻ. ࢝ ିࢃࢀ. ቕሻǣ ൌ ࡶ ሺࢃ࢞ࢀǡ ࢝ǡ ሻǤ. 3c ' c= c& c>!0# c ࢃࢀିሺିሻ ࢃ ࢃࢀି ǥ ǥ ǥ ቔ ࢎቕ ࢝ ࢝ି ࢝ିሺିሻ. 5 ሺࢃ࢞ࢀǡ ࢝ǡ ሻ ඌ ቔ. ሺௐ்ିሻሺ௪ିሻ. c F ൌ ቀܹܶ െ ݈ǡ ቔሺ௪ିሻ మ. ିሺௐ்ିሻሺିሻ. ǥ ቕඐ ǣ ൌ ࡶ ሺࢃ࢞ࢀǡ ࢝ǡ ሻǤ. ቕቁǤ݈ cc cͳ ݈ ݇ െ ͳǡc. ccሺ ݓെ ݈ ሻଶ ሺܹܶ െ ݈ ሻሺ݇ െ ݈ ሻǤc c. c. cc c.
(280) " ñ ݊ ൌ ܹܶ ݅ܬǡ ݅ אሼܣǡ ܤǡ ܥሽ ሺܹܶǡ ݓǡ ݇ሻPP
(281) ሺܹܶǡ ݓǡ ݇ሻPP
(282) ࡶ܅ሺ܂܅ǡ ܟǡ ܓሻ ࡶ ሺ܂ܠ܅ǡ ܟǡ ܓሻ ࡶ܅ሺ܂܅ǡ ܟǡ ܓሻ ሺ ܅െ ሻ. ݅ܬሺܹܶǡ ݓǡ ݇ሻ PP
(283)
(284) ñ PP
(285)
(286) ࢫ · c 4( c $$c c &c c c c c c c c cܹ ൌ ݂ ሺݐሻǡ Ͳ ൏ ݐ൏ ܶ െ ͳǡ Ͳ ൏. ܹ ൏ ܹ ൌ ͳǡc cc c
(287) c c ( ݐൌ ݂ሺߣሻc. c c c c. ܵఛ c c ܵఛ ሺȈǡ ݂ሺȈሻሻ c c c ߬c
(288) c c c c c ൫Ȉǡ ݂ ሺȈሻ൯c
(289) c c. c%ccc -c c ccc ccc
(290) c$$cc :c
(291) c c ݇ c. c. ' c c ܵఛ ൫ݔǡ ݂ ሺ ݔሻ൯ ൌ ൫ݔǡ ݂ ሺ ݔሻ൯ c c %c ݇ c
(292) c c c ݂ c c c ݃ c c. c݂ ് ݃ c ߬ ് Ͳc c. 88 9c c c
(293) c c c ò ሺ ݍሻ c c 4( c $$-c c c
(294) c. c c ò ሺݍሻc c&c ܿሺݐሻc c. ሺ௫ሻ. ሺ௫ሻ. c c
(295)
(296) c >c. 3c ݂ ݃ ݐ 3c ݂݃ 3c. ሺ௫ሻ. ሺ௫ሻ. ് ܽǡ ܽ . 3c ݂ . c. cc c.
(297) Jܿሺ ሻ s gven as ࢉሺ ሻ. ି ቊ ା. ૠ. ି Τ. . ૠ. The above condons guaraneed ha each funcon s couned on
(298) y once. Nሺ௫ሻ.
(299)
(300) . ሺ௫ሻ. ܿݏ . Two. ܿ ܿ e
(301) emens Õ Õ are equa
(302) provded here exs an e
(303) emen such haÕ Õ . ݀ ݀. " #ሺ ሻ .
(304) ሺ ሻ ߁ . . . . .
(305). ሺ ሻ
(306) . !. !
(307)
(308)
(309) ! .
(310) ߁ ሾ ሾ
(311) !
(312) ሺ ሻ ሺ ሻ I fo
(313)
(314) ows ha he e
(315) emens of. ሺሻ. can be arranged so as o form an orb of. s e (q +
(316) > ۾ ሺܙሻ ܑ Õ ܑ ! ܙǢ Õ
(317) ۾ ሺܙሻ ቅ . À
(318) À. . . À Þ À À e " # " $, and #% funcons. Nሺఒሻ. ሺఒሻ. some subse of ሺ$ሻ of s e#. Here we consder raona
(319). mappng wave
(320) engh no me. Assocae o each me s
(321) o , he h e
(322) emen of. a cyc
(323) c represenaon of. ሺ ሺ$ሻሻ. e us defne>. Where, gven a raona
(324) funcon, he operaor ܰ dvdes ou he common facors beween numeraor and denomnaor and n addon, sca
(325) es he wo so as o make he numeraor monc [9].. c. cc c.
(326) onsderng # and are re
(327) ave
(328) y prme, and. & , s obvous ha he operaor ܰresu
(329) s. n some raona
(330) funcon whch sasfes he condons of secon 5. We need o dscard a
(331)
(332) raona
(333) funcons whch are of he form>. We noe ha such a raona
(334) funcon doesn' exs snce means ha for ceran for some " ݇ " $ whch s mpossb
(335) e by Theorem 2. Amongs he funcons sasfyng he condons of secon 5, we have dscarded a
(336)
(337) consan funcons, bu as hs sep s unnecessary here, we can add hem back. For any " ݇ " $, we dec
(338) are wo raona
(339) funcons. o be. equva
(340) en. Then he dfferen code marces correspond o choosng precse
(341) y one po
(342) ynoma
(343). from each equva
(344) ence c
(345) ass. For each po
(346) ynoma
(347) #ሺ ሻ he ሺܹ'ሻ code array. wh s e,. ሺሻᇱ ା. and݇ (݇ᇱ. s gven by. . Ths resu
(348) s n aሺܹ ሺ$ ሻ ܹ (݇ᇱ ሻ 2-D OO ܹ. $). À ? À À e 'ȁሺ$ ሻ and *% ሺ$ሻhave mu
(349) p
(350) cave order'. Take he wave
(351) enghs as. ሺ ሺ$ሻሻ(he order of e
(352) emens doesn' maer). Here we consder raona
(353) funcons. mappng me no wave
(354) engh. e us assocae o me s
(355) o+, he e
(356) emen* . We defne wo raona
(357). funcons. n for some. whch raona
(358) funcons s compued n [9] as>. c. . o. be. equva
(359) en. . Frs dscard a
(360)
(361) raona
(362) funcons. f. Nሺ௫ሻ. ሺ௫ሻ. sasfy= for & . The number of remanng .. cc c.
(363) hoosng one funcon from each of he remanng equva
(364) ence c
(365) asses and assocang. o. ,. he. ሺܹ'ሻ. code. by. array.
(366) eng. where. and. resu
(367) s n a. 2-D OO of s e. !
(368) À We verfy he auo-corre
(369) aon J cross-corre
(370) aon properes of 2-D OO codes by usng ATA5 .The wo. codes are chosen form he 8 codes shown ear
(371) er n hs repor.. Smu
(372) aon s done wo pars 1.corconv.m o fnd convo
(373) uon. J 2.ocdma.m o generae code, corre
(374) ae J p
(375) o hem. ¬c. Ths s represened as marx as code1= . . . . . . -,. code5= . . . . . . -,. "
(376) !
(377)
(378)
(379) ! #
(380)
(381)
(382)
(383)
(384)
(385)
(386)
(387) ! !
(388) . c. cc c.
(389) !
(390) À
(391) . c. .
(392) À À ÀÀ ÀÀÀ cc c.
(393) "
(394) . !#c?c 5C c D?;c E'9$CFcc%
(395) c;cc E G c c cCH?+c ECcc&c cc7 E Gc c c7 4 I"c cc ? EGc c c754 I"c cc5?5EGc c cCI
(396) c cc ?! cGc7 C(
(397) " #Gc c cCI
(398) 5 c cc5?!5cGc7 C(
(399) 5 " #Gc c c?c ">C Gc 5c?c5">C Gc .
(400) -&
(401) c?c J5EGc .
(402) -&
(403) c?c
(404)
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(641) p £ p F. R. K. Chung, J. A. Salehi, and V. K. Wei, "ptical orthogonal codes: Design, analysis, and applications,"p pp
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(643) p vol. 36, pp. 866-873, July 1990. ë p å. A. Nguyen, L. Gyorfi, and J. L. Massey, "Constructions of binary constant- weight cyclic codes and cyclically permutable codes,"p pp
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(650) vol. 185, pp. 201-219, 1998. V pR. Fuji-Hara and Y. Miao, "ptical orthogonal codes: Their bounds and new optimal constructions,"p pp
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(654) vol. 14, pp. 256-266, 2001. VÔ pY. Tang and J. Yin, 'The combinatorial construction for a class of optimal optical orthogonal codes,"p
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(664) V pY.. Chang, R. Fuji-Hara, and Y. Miao, "Combinatorial constructions of optimal optical orthogonal codes with weight 4,"p pp
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